Papers and preprints

We introduce and study the main properties of certain Dedekind sums attached to a totally real number field.
These are not rational numbers but rather real-analytic functions. They arise in the transformation formula of an ad-hoc analog of the logarithm of the Dedekind eta function for the Hilbert modular group.

- arXiv numbermath.NT/0405262 (original and expanded version of the above): (sommes_arxiv)

We describe a conjectural construction (in the spirit of Hilbert’s 12th problem) of units
in abelian extensions of certain base fields which are neither totally real nor CM. These
base fields are quadratic extensions with exactly one complex place of a totally real number
field F,
and are referred to as Almost Totally Real (ATR) extensions. Our construction involves
certain null-homologous topological cycles on the Hilbert modular variety attached to
F. The special
units are the images of these cycles under a map defined by integration of weight two Eisenstein
series on GL2(F).
This map is formally analogous to the higher Abel–Jacobi maps that arise in the theory
of algebraic cycles.
We show that our conjecture is compatible with Stark’s conjecture
for ATR extensions ; it is, however, a genuine strengthening of Stark’s conjecture in this
context since it gives an analytic formula for the arguments of the Stark units and not
just their absolute values. The last section provides numerical evidences for our conjecture.

We define an integral version of Sczech's Eisenstein cocycle on GLn by smoothing at a prime ell. As a result we obtain a new proof of the integrality of the values at nonpositive integers of the smoothed partial zeta functions associated to ray class extensions of totally real fields. We also obtain a new construction of the p-adic L-functions associated to these extensions. Our cohomological construction allows for a study of the leading term of these p-adic L-functions at s=0.
We apply Spiess's formalism to prove that the order of vanishing at s=0 is at least equal to the expected one, as conjectured by Gross. This result was already known from Wiles' proof of the Iwasawa Main Conjecture.

We define a cocycle on Gln using Shintani's method.
This cocycle is closely related to cocycles defined earlier by Solomon and Hill, but
differs in that the cocycle property is achieved through the introduction of an auxiliary perturbation vector Q.
As a corollary of our result we obtain a new proof of a theorem of Diaz y Diaz and Friedman on signed fundamental
domains, and give a cohomological reformulation of Shintani's proof of the Klingen--Siegel rationality theorem
on partial zeta functions of totally real fields.
Next we prove that the cohomology class represented by our Shintani
cocycle is essentially equal to that represented by the Eisenstein cocycle defined by Sczech. This generalizes
a result of Sczech and Solomon in the case n=2. It implies that the formulae for values of partial zeta functions
arising from Shintani's method and from Sczech's method are identical.
Finally we introduce an integral version of our Shintani cocycle by smoothing at an auxiliary prime ell.
Applying the formalism of the first paper in this series, we prove that certain specializations of the smoothed class yield the p-adic L-functions
of totally real fields. Combining our cohomological construction with a theorem of Spiess, we
show that the order of vanishing of these p-adic L-functions is at least as large as the
expected one.

Lalin, Rodrigue and Rogers recently introduced the secant zeta function,
as an analog of the Lerch-Berndt-Arakawa cotan zeta function. These two zeta functions are Dirichlet series attached to a real quadratic irrational number α.
They conjectured that, similarly to the cotan zeta function case, the values at a specific range of negative integers of the secant Dirichlet series are algebraic numbers that belong to the field generated by α.
In this short note we provide two different proofs of their conjecture.

This article introduces a set of recently discovered lecture notes from the last course of Leopold Kronecker, delivered a few weeks before his death in December 1891. The notes, written by F. von Dalwigk, elaborate on the late recognition by Kronecker of the importance of the ``Eisenstein summation process'', invented by the ``companion of his youth'', in order to deal with conditionally convergent series known today as Eisenstein series.
We take this opportunity to give a brief update of the well known book by André Weil (1976) that brought these results of
Eisenstein and Kronecker back to light.
We believe that Eisenstein's approach to the theory of elliptic functions was in fact a very important part of Kronecker's planned proof of his visionary ``Jugendtraum''.

In this paper, we explicitly construct harmonic Maass forms that map to the weight one Theta series associated by Hecke to odd ray class group characters of real quadratic fields. From this construction, we give precise arithmetic information contained in the Fourier coefficients of the holomorphic part of the harmonic Maass form, establishing the main part of a conjecture of the second author.